Ratio Scale

Ratio Scale for Costs, Survival Time and Utilities

A ratio scale is the highest of the four levels in Stevens's 1946 classification of measurement scale types, and the only one whose zero marks a genuine absence of the quantity being measured. Most of the quantities on the cost side of an economic evaluation are ratio-scale, including costs in money, counts of resource use such as GP visits, length of stay and survival time. On the outcome side the position is less settled, because the utility scale used to build QALYs is anchored at dead and full health and includes values below zero, and health economists disagree over whether its zero is a true zero. That disagreement matters for practice, since moving the zero of the utility scale changes the QALY gain of any treatment that extends life.

Equal intervals and a true zero

A ratio scale has equal intervals, like an interval scale, and in addition a zero that marks the absence of the quantity rather than an arbitrary point. Money, counts of events and lengths of time all behave this way in health economic data. Because the zero is fixed, the only permissible transformation is a change of unit,

$$x' = b x, \quad b > 0$$

where $x$ is the original value, $x'$ is the value in the new unit and $b$ is the conversion factor. Under this transformation the ratio of any two values is unchanged, so a statement that one quantity is twice another is true in every unit or false in every unit. An interval scaling also allows a constant to be added, which preserves differences but not ratios.

The ratio-scale quantities common in health economics share a natural zero. No resource used means zero cost, no admissions means a count of zero, and no time lived means a duration of zero. A nominal scale code or an ordinal level has no such zero, and neither does a temperature in degrees Celsius, the textbook case of an interval scale.

Costs and resource use on a ratio scale

A cost is a quantity of resource multiplied by a unit price, and both factors are ratio-scale, so the product is too. A course of treatment costing GBP 3,000 uses twice the resources, valued at the same prices, as one costing GBP 1,500. Converting costs into another currency or uplifting them to a later price year multiplies every value by the same factor, which is the permissible transformation of a ratio scale, so relative statements about costs survive both adjustments.

Incremental costs can be negative when a treatment saves money, but that does not weaken the scale, because an incremental cost is a difference between two ratio-scale totals. The incremental cost-effectiveness ratio divides one difference by another, the extra cost by the extra health effect, and a change of currency rescales the result by the exchange rate. In the worked example GBP 30,000 per QALY becomes EUR 34,500 per QALY at 1.15 euros per pound, and what survives the conversion is the comparison with a threshold expressed in the same currency. Moving the origin of the utility scale is a different matter, because it changes the QALY gain itself whenever survival differs between arms.

Analysing ratio-scale costs from trials

The ratio level also explains why budget holders care about the arithmetic mean of skewed patient costs. Total spending equals the mean cost multiplied by the number of patients, a statement that depends on costs adding up as amounts, which they do because zero cost is a real zero. Patient-level costs in trials are usually right-skewed, with a few very expensive patients, and that shape tempts analysts towards medians or log transformations.

A log transformation turns ratios into differences, so a comparison of mean log costs estimates the ratio of geometric means between arms. That is a legitimate ratio-scale statement, but it answers a different question from the one a payer asks, which concerns the difference in arithmetic mean cost per patient. Barber and Thompson, reviewing cost analyses in published randomised trials, pointed to the bootstrap as a way of comparing arithmetic means that needs no assumption about the shape of the cost distribution. The level of measurement tells the analyst which statistics are permissible, and the decision tells the analyst which of those statistics is needed.

Life-years and the duration scale

Survival time is the second ratio-scale quantity in a QALY. Prieto and Sacristán stated the contrast directly, writing that life-years are expressed in a ratio scale with a true zero, while utility is an interval scale on which 0 is an arbitrary value for death. They argued that the QALY needs both scales to share ratio properties and treated the interval status of conventional utility as a defect of the QALY.

Roudijk, Donders and Stalmeier approached the question through indifference between health episodes. They argued that indifference between two episodes is not preserved under changes of the origin of the duration scale, so ratio properties are needed for that scale, and their review concluded that the health-utility and duration scales both have ratio scale properties and that dead should be assigned 0.

Whether the utility scale has a true zero

Value sets used to calculate QALYs fix full health at 1 and dead at 0. Those anchors give the scale its unit and its origin, but they do not settle whether the origin is a true zero in Stevens's sense. Prieto and Sacristán and Roudijk and colleagues agree that the QALY model needs a utility scale with ratio properties and differ on whether conventional utilities have them. For Prieto and Sacristán, dead sits at zero by convention and utility remains interval, so a statement that a health state utility value of 0.8 is twice as good as 0.4 has no meaning and the QALY inherits a defect. Roudijk and colleagues conclude that dead should be assigned 0 and that the health-utility scale then has the ratio properties the model uses.

The time trade-off shows why a ratio reading is attractive. A respondent indifferent between 10 years in a state and 7 years in full health gives the state a value of 7 / 10 = 0.7, a ratio of two durations, and that ratio means what it says only if immediate death is worth 0.

Values below zero are compatible with either reading. Torrance's review of utility measurement for economic appraisal covered states both better and worse than death, and Roudijk and colleagues noted that the measurement method itself affects whether a state is judged better or worse than dead. In the England EQ-5D-5L value set reported by Devlin and colleagues in 2018, the lowest value is −0.285 for extreme problems on all dimensions. A negative value means a state is valued below dead, which requires dead to be a fixed reference point but does not by itself show that 0 is an absence of value.

Worked example with an illustrative shift in the utility zero

All numbers in this example are illustrative. A new treatment A gives 4 years of survival at a utility of 0.5, and the comparator B gives 2 years at a utility of 0.9. Treatment A costs GBP 18,000 and treatment B costs GBP 12,000.

Step 1 (QALYs with dead at zero). Treatment A gives 4 × 0.5 = 2.0 QALYs and treatment B gives 2 × 0.9 = 1.8 QALYs. The QALY gain is 2.0 minus 1.8, which is 0.2.

Step 2 (incremental cost and ICER). The incremental cost is GBP 18,000 minus GBP 12,000, which is GBP 6,000. The ICER is 6,000 / 0.2 = 30,000, so GBP 30,000 per QALY gained.

Step 3 (moving the origin of the utility scale). An interval scale permits adding a constant, and here 0.2 is added to every utility while dead keeps a QALY weight of 0. Treatment A now gives 4 × 0.7 = 2.8 QALYs and treatment B gives 2 × 1.1 = 2.2 QALYs. The QALY gain becomes 0.6 and the ICER becomes 6,000 / 0.6 = 10,000, so GBP 10,000 per QALY gained.

Step 4 (equal survival in both arms). If both treatments gave 3 years, at utilities of 0.6 and 0.5, the gain would be 3 × 0.1 = 0.3 QALYs. After adding 0.2, the arms give 3 × 0.8 = 2.4 and 3 × 0.7 = 2.1 QALYs, and the gain is still 0.3.

Step 5 (changing the currency). At an illustrative rate of 1.15 euros per pound, treatment A costs 18,000 × 1.15 = 20,700 and treatment B costs 12,000 × 1.15 = 13,800, so EUR 20,700 and EUR 13,800. The cost ratio is 18,000 / 12,000 = 1.5 in pounds and 20,700 / 13,800 = 1.5 in euros. The incremental cost becomes EUR 6,900 and the ICER becomes 6,900 / 0.2 = 34,500, so EUR 34,500 per QALY, which compares with a threshold converted at the same rate exactly as before.

The shift in step 3 is harmless to an interval scale but triples the QALY gain and cuts the ICER to a third, because the extra 2 years of survival in arm A are now weighted more heavily. Step 4 shows that the origin does not matter when survival is equal, which is why the dispute over the zero bites hardest on life-extending treatments. Step 5 shows the contrast with costs, where the permissible change of unit leaves cost ratios as they were and rescales the ICER without changing its comparison with a threshold in the same currency.

Ratio statements in reporting

A ratio scale supports relative statements that other levels do not. A treatment can be reported as costing 50 per cent more than its comparator, or a length of stay as half as long, and the statement holds in any currency or unit of time. Geometric means and coefficients of variation are also meaningful only for ratio-scale data.

Relative statements about utilities need more care. A fall in mean utility from 0.8 to 0.4 is a loss of 0.4 on the scale, but describing it as halving quality of life assumes that 0 is a true zero, and the same fall described against a different origin would be a different percentage. QALY gains are differences and are reported as such in a cost-utility analysis, which avoids the problem.

Boundaries with ratio scaling and relative measures

Ratio scaling is a valuation procedure in which respondents state how many times better or worse one state is than another. It aims to produce values on a ratio scale, but the level of the resulting numbers depends on whether respondents can make such judgements consistently, so the method and the scale type are separate ideas.

Relative measures of effect, such as a relative risk or a hazard ratio, are ratios between two groups. They compare probabilities or rates, which have a fixed zero, so their ratio form is meaningful, but they are effect measures and not a level of measurement.

Limitations and common misreadings

The ratio level of costs depends on what is counted. A cost of zero in a dataset may reflect missing records rather than no resource use, and structural zeros of that kind need separate treatment before ratio statements are made.

1. Negative costs show costs are not ratio-scale. Negative values arise in differences, not in the total cost of an arm.
2. Utilities anchored at 0 and 1 are automatically ratio-scale. The anchors fix the origin by convention, and whether conventional utilities have the ratio properties the QALY needs is the point in dispute.
3. QALY ratios are as safe as cost ratios. A ratio of QALYs between arms depends on where dead sits, so a QALY gain is reported as a difference.

Sources

• Stevens SS. On the theory of scales of measurement. Science. 1946;103(2684):677-680. https://doi.org/10.1126/science.103.2684.677
• Prieto L, Sacristán JA. Problems and solutions in calculating quality-adjusted life years (QALYs). Health and Quality of Life Outcomes. 2003;1:80. https://doi.org/10.1186/1477-7525-1-80
• Roudijk B, Donders ART, Stalmeier PFM. Setting dead at zero: applying scale properties to the QALY model. Medical Decision Making. 2018;38(6):627-634. https://doi.org/10.1177/0272989X18765184
• Torrance GW. Measurement of health state utilities for economic appraisal. Journal of Health Economics. 1986;5(1):1-30. https://doi.org/10.1016/0167-6296(86)90020-2
• Devlin NJ, Shah KK, Feng Y, Mulhern B, van Hout B. Valuing health-related quality of life: an EQ-5D-5L value set for England. Health Economics. 2018;27(1):7-22. https://doi.org/10.1002/hec.3564
• Barber JA, Thompson SG. Analysis and interpretation of cost data in randomised controlled trials: review of published studies. BMJ. 1998;317(7167):1195-1200. https://doi.org/10.1136/bmj.317.7167.1195